We use recent astrophysical and local tests of the stability of the fine-structure constant, α, to constrain a particular phenomenological but physically motivated class of models in which the dark energy equation of state can undergo a rapid transition at low redshifts, perhaps associated with the onset of the acceleration phase. We set constraints on the phenomenological parameters describing such possible transitions, in particular improving previous constraints (which used only cosmological data) on the present-day value of the dark energy equation of state in these models. We specifically quantify how these constraints are improved by the addition of the α measurements. We find no evidence for a transition associated with the onset of acceleration. In this model the α measurements lead to a bound on Weak Equivalence Principle violations of η < 4 ×10^-15 (at 68.3% confidence level), improving on the recent MICROSCOPE bound by about a factor of three.